From Algorithm  we can see that LLP requires predictions for the next gradient \(\tilde{c}_{t}\!=\!\nabla \tilde{f}_t(\tilde{x}_t)\) , next cost function \(\tilde{g}_t(\cdot )\) , and next constraint point \(\tilde{g}_t(\tilde{x}_t)\) . It is important here to note the timing of these predictions. Namely, updating \(x_t\)  requires \(\tilde{c}_t\)  and \(\tilde{g}_t(\cdot )\)  and access to regularizers \(r_{0:t-1}\)  which are calculated using the prediction errors up to slot \(t-1\) . Knowledge of \(\tilde{c}_t\)  is the standard prediction that all prior works employ, e.g., see [1]}, [2]} and references therein. On the other hand, since we have not linearized the constraint function, the respective predictions involve function \(\tilde{g}_t(\cdot )\)  and its next-round value \(\tilde{g}_t(\tilde{x}_t)\) . In Section  we present a version of LLP where we linearize the constraints and hence use only gradient predictions for the constraints – similarly to cost functions.
